This paper deals with first passage heights of sums of renewal sequences, random walks, and Lévy processes. We prove that the joint age and excess (and therefore, the current life) stationary distributions of these heights are stochastically increasing (in the usual first-order sense) in the passage levels. As a preliminary tool, which is also of independent interest, a new decomposition of the stationary excess distribution, as a convolution of two other distributions, is developed. As a consequence of these results, certain monotonicity results are concluded for ratios involving convex functions. This paper is motivated by problems related to control of queues with removable servers which model single-machine produce-to-order manufacturing systems. Applications to these problems are provided.
"Stochastic monotonicity for stationary recurrence times of first passage heights." Ann. Appl. Probab. 7 (2) 326 - 339, May 1997. https://doi.org/10.1214/aoap/1034625334